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These are the user uploaded subtitles that are being translated: 1 00:00:00,540 --> 00:00:05,010 Today we're going to be talking about how to determine whether or not a geometric series is convergent 2 00:00:05,100 --> 00:00:06,320 or divergent. 3 00:00:06,540 --> 00:00:12,080 In other words we're going to be using the geometric series test for convergence and the series. 4 00:00:12,120 --> 00:00:17,040 In this particular video that we're going to be testing is the infinite sum from Ed equals one to infinity 5 00:00:17,460 --> 00:00:24,910 of this series here which is negative 3 Race to the end minus one power divided by 4 to the power. 6 00:00:24,960 --> 00:00:31,080 Now as a reminder I have the geometric series convergence test written here or at least part of it. 7 00:00:31,080 --> 00:00:34,230 It says that the infinite sum from Ed equals one to infinity. 8 00:00:34,230 --> 00:00:41,310 Of this type of series which is a geometric series a times are raised to the end minus one power that 9 00:00:41,310 --> 00:00:42,860 that series is convergent. 10 00:00:42,870 --> 00:00:48,840 If the absolute value of r is less than 1 and divergent if the absolute value of r is greater than or 11 00:00:48,900 --> 00:00:50,420 equal to 1. 12 00:00:50,440 --> 00:00:57,570 Now the easiest way to use this geometric series convergence test is to first expand this series here 13 00:00:57,570 --> 00:01:01,280 that were given in the definition into a series. 14 00:01:01,350 --> 00:01:06,450 And the way that we're going to do that is by plugging in values of and starting with N equals 1. 15 00:01:06,630 --> 00:01:11,520 We'll keep plugging in higher and higher values of and and just get the first couple of terms of this 16 00:01:11,940 --> 00:01:18,420 series then it'll become much more apparent to us what kind of series a geometric series actually is. 17 00:01:18,420 --> 00:01:23,330 So we'll go ahead and just say that this is going to be roughly equal to here. 18 00:01:23,460 --> 00:01:30,420 When we plug in and equals 1 to this series here we at a times are to the one minus one power one minus 19 00:01:30,420 --> 00:01:35,150 one is ZERO are raised to the zero power is 1. 20 00:01:35,160 --> 00:01:40,070 So we essentially have here a times one and we just get a. 21 00:01:40,290 --> 00:01:46,800 Ok then we're going to add to that whatever we get when we plug in and equal to 2 minus 1 gives us once 22 00:01:46,830 --> 00:01:50,510 we get a times are to the first power or just a r. 23 00:01:50,520 --> 00:01:59,600 So plus a R when we plug in unequals 3 we get 3 minus 1 which is 2 or a r squared. 24 00:01:59,610 --> 00:02:03,140 So plus a r squared. 25 00:02:03,270 --> 00:02:09,510 And if we kept going here what we'd see is that we get a r cubed a r to the fourth etc.. 26 00:02:09,550 --> 00:02:12,040 I mean just going to say Dot dot dot here. 27 00:02:12,360 --> 00:02:12,890 OK. 28 00:02:13,020 --> 00:02:18,210 So this is the expanded form here of a geometric series. 29 00:02:18,210 --> 00:02:24,840 Now what we can do what's really interesting here is that we can factor out the value from this series. 30 00:02:24,840 --> 00:02:28,060 Notice that each term in the series is multiplied by a. 31 00:02:28,320 --> 00:02:34,890 So when we factor out a like this what we get is one right a times one is just a that gives us our first 32 00:02:34,890 --> 00:02:42,300 term plus our when we factor in the second term a times are gives us a r here and we keep factoring 33 00:02:42,300 --> 00:02:48,750 out we just get plus R squared plus R Q plus dot dot dot. 34 00:02:49,170 --> 00:02:51,750 We get to the fourth R the fifth etc.. 35 00:02:51,900 --> 00:02:55,300 So here's our geometric series now with a factored out. 36 00:02:55,380 --> 00:03:01,770 The reason it's convenient to understand that this is the expanded form of a geometric series is because 37 00:03:01,830 --> 00:03:10,110 we can now plug in values of an AND equals 1 2 3 etc. into our series appear to see if it matches this 38 00:03:10,110 --> 00:03:10,760 form. 39 00:03:10,860 --> 00:03:15,800 A Times 1 plus R plus R-squared plus argued to see if it matches that form. 40 00:03:15,810 --> 00:03:19,710 If it does match that form then we can tell that it's a geometric series. 41 00:03:19,710 --> 00:03:26,340 We can easily tell it's a geometric series and we can also easily identify values of a and r which is 42 00:03:26,340 --> 00:03:31,290 what we're interested in not only for the convergence test but also for finding the sum of a geometric 43 00:03:31,290 --> 00:03:32,040 series. 44 00:03:32,040 --> 00:03:34,130 If we need to do that as well. 45 00:03:34,150 --> 00:03:40,010 So again the easiest way to determine whether or not we actually have a geometric series here is just 46 00:03:40,010 --> 00:03:41,600 a plug in the first several terms. 47 00:03:41,700 --> 00:03:43,680 So let's go ahead and do that. 48 00:03:43,680 --> 00:03:47,170 Here are series and we'll just go ahead and put it in brackets. 49 00:03:47,310 --> 00:03:53,280 When we plug and equals one into this series here we'll get in the numerator one minus one which is 50 00:03:53,340 --> 00:03:54,120 zero. 51 00:03:54,120 --> 00:03:59,940 Negative 3 raise to the zero power as just once we get 1 in the numerator in the denominator when we 52 00:03:59,940 --> 00:04:03,670 plug in one we get four to the first power which is just 4. 53 00:04:03,690 --> 00:04:09,060 So our first term in our series is one fourth that we're going to add to that whatever we get only plug 54 00:04:09,120 --> 00:04:10,620 in 2. 55 00:04:10,650 --> 00:04:16,890 So I plug in to we get to minus 1 which is 1 negative 3 raise to the first power is just negative 3 56 00:04:16,890 --> 00:04:23,490 so negative 3 divided by four raised to the second power of four squared here which is 16. 57 00:04:23,490 --> 00:04:27,300 So we get negative 3 over 16 for our second term. 58 00:04:27,300 --> 00:04:29,290 Now let's just do two more really quickly. 59 00:04:29,460 --> 00:04:34,940 When we plug in three we get three minus one which is to negative three squared is nine. 60 00:04:34,950 --> 00:04:39,740 So we get positive nine divided by four cubes which is 64. 61 00:04:39,750 --> 00:04:41,490 So nine over 64. 62 00:04:41,790 --> 00:04:48,900 And if we plug in an equals four we get four minus one which has three negative three cubed is negative 63 00:04:48,960 --> 00:04:49,870 27. 64 00:04:49,920 --> 00:04:57,210 So negative 27 divided by four to the fourth power which is two hundred fifty six and we could just 65 00:04:57,210 --> 00:05:02,520 keep going but we'll go ahead and say dot dot dot like this and close our series. 66 00:05:02,520 --> 00:05:02,950 OK. 67 00:05:03,000 --> 00:05:05,840 So these are the first several terms of our series. 68 00:05:05,940 --> 00:05:10,440 Now in order to determine whether or not this is a geometric series once you've got these terms right 69 00:05:10,440 --> 00:05:15,960 now what you always want to try to do is factor out the value of our first term. 70 00:05:15,960 --> 00:05:21,210 Remember here we had our first term a and we tried to factor out a. 71 00:05:21,360 --> 00:05:28,200 So in our case we have our first term one fourth and we want to try to factor out one fourth. 72 00:05:28,200 --> 00:05:35,040 So what we're going to do is say one fourth and then multiply this by all these terms in the inside 73 00:05:35,040 --> 00:05:35,470 here. 74 00:05:35,640 --> 00:05:40,890 So one fourth obviously the first term will be multiplied by 1 to get one fourth again. 75 00:05:40,950 --> 00:05:45,070 Now what do we have to multiply by one fourth in order to get negative 3 over 16. 76 00:05:45,090 --> 00:05:51,780 Well we have to multiply that by negative 3 over 4 right now if we take a negative three fourths and 77 00:05:51,780 --> 00:05:53,090 we multiply it by one fourth. 78 00:05:53,100 --> 00:05:58,320 We get negative three in the numerator and 16 in the denominator which is what we have here for our 79 00:05:58,320 --> 00:06:05,070 second term negative 3 over 16 where we have to multiply by one fourth in order to get nine over 64. 80 00:06:05,100 --> 00:06:07,460 Well we have to multiply that by nine. 81 00:06:07,650 --> 00:06:13,350 And then in the denominator we have to multiply that by 16 and we could keep going here we have to multiply 82 00:06:13,350 --> 00:06:17,560 this by negative twenty seven over 60. 83 00:06:17,580 --> 00:06:19,800 For me it's a plus dot dot dot. 84 00:06:20,070 --> 00:06:24,280 And this would be our series with the first term. 85 00:06:24,300 --> 00:06:25,770 One fourth factor it out. 86 00:06:25,770 --> 00:06:30,900 Now all we can do to verify that this is a geometric series is make sure that we have it in the same 87 00:06:30,900 --> 00:06:34,690 form as our series here and identify a and are. 88 00:06:34,860 --> 00:06:40,170 Well theoretically if this is a geometric series we already factored out our value of a we're already 89 00:06:40,170 --> 00:06:46,830 assuming that a is one fourth because it was our first term we got we went ahead and factor that out. 90 00:06:46,860 --> 00:06:52,590 Now our is always going to be our second term here and we want to make sure that we always include a 91 00:06:52,590 --> 00:06:54,360 negative sign if there is one. 92 00:06:54,360 --> 00:07:00,360 So this is going to be our value of our including the sign in front of it whether it's positive or negative. 93 00:07:00,390 --> 00:07:06,580 So in our case that would be negative three fourths would be our value of our. 94 00:07:06,580 --> 00:07:12,690 Now notice that all we have to do is make sure that the terms following are are in fact are squared 95 00:07:12,690 --> 00:07:14,890 are cubed art of the fourth et cetera. 96 00:07:15,180 --> 00:07:20,910 So what we want to do is go ahead and break these down to see if we in fact say that if this is our 97 00:07:20,970 --> 00:07:22,670 then this is our squared. 98 00:07:22,680 --> 00:07:24,910 This is our cubes et cetera. 99 00:07:25,050 --> 00:07:31,850 So we'll have one fourth multiplied by one minus three fourths. 100 00:07:31,860 --> 00:07:34,820 And now what we're going to try to do is get three fourths squared. 101 00:07:34,830 --> 00:07:42,380 And obviously we can see that that is going to be the case we're going to say plus three fourths squared 102 00:07:42,400 --> 00:07:47,090 here because we can see that if we square three we get nine if we square four we get 16. 103 00:07:47,190 --> 00:07:49,890 That's nine sixteenths which was r r squared term. 104 00:07:49,920 --> 00:07:54,780 And in fact what we can do is because we know that our value for our is negative three fourths we can 105 00:07:54,780 --> 00:07:57,620 go ahead and add in our negative rate here. 106 00:07:57,750 --> 00:08:03,950 Negative three four square gives a positive nine 16th which is what we need here for our fourth term 107 00:08:03,960 --> 00:08:13,240 this 27 or 64 we can go ahead and say plus and then negative three fourths cubed. 108 00:08:13,340 --> 00:08:18,190 Because when we cube the negative will still end up with a negative right negative one race to the third 109 00:08:18,190 --> 00:08:23,860 power still negative one we get this negative sign out in front and three cubes is 27 and four cubed 110 00:08:23,890 --> 00:08:25,220 is sixty four. 111 00:08:25,270 --> 00:08:26,300 So we're getting that value. 112 00:08:26,320 --> 00:08:31,040 OK so we've proven we've shown that negative three fourths here. 113 00:08:31,060 --> 00:08:35,730 This second term inside this series is in fact our value of our. 114 00:08:35,740 --> 00:08:39,250 And that if we keep going with that we get R-squared argued of the fourth. 115 00:08:39,250 --> 00:08:45,520 So we've shown that this is a geometric series because we've matched the form of our series to the definition 116 00:08:45,880 --> 00:08:47,920 of a geometric series here. 117 00:08:48,100 --> 00:08:54,220 And we know that one fourth is equal to a and then negative three fourths is equal to r. 118 00:08:54,550 --> 00:08:59,530 So now when it comes to the convergence test all we need to do is look at our right we're taking the 119 00:08:59,530 --> 00:09:00,930 absolute value of our. 120 00:09:00,970 --> 00:09:08,410 So we want the absolute value of negative three fourths which of course is equal to positive three fourths. 121 00:09:08,410 --> 00:09:12,430 Is this less than or greater than or equal to 1. 122 00:09:12,430 --> 00:09:14,080 Of course it's less than 1. 123 00:09:14,140 --> 00:09:20,170 So because three fourths is less than 1 We know that the series is convergence so we'll say that it 124 00:09:20,170 --> 00:09:25,120 is a convergent series by the geometric series tests. 125 00:09:25,180 --> 00:09:30,590 If this value had been greater than or equal to one we would know that the series is divergent. 126 00:09:30,820 --> 00:09:36,520 And now as a quick note about the sum of the series once you have a and are finding the sum of the series 127 00:09:36,610 --> 00:09:38,050 is very easy. 128 00:09:38,050 --> 00:09:45,100 The sum of a geometric series is always just a divided by 1 minus R and since we already have values 129 00:09:45,100 --> 00:09:46,930 of a n r that's very easy. 130 00:09:46,930 --> 00:09:54,640 So our value of a is one fourth our value for our is negative three fourths so you get 1 minus negative 131 00:09:54,640 --> 00:09:58,620 three fourths or 1 plus three fourths. 132 00:09:58,630 --> 00:10:01,080 And now we just need to simplify this to find the sum. 133 00:10:01,120 --> 00:10:06,610 The way that you want to do this is by finding a common denominator within the denominator so this one 134 00:10:06,610 --> 00:10:12,670 here becomes for over 4 to find a common denominator with this three fourths. 135 00:10:12,670 --> 00:10:18,210 That leads us to one fourth divided by seven fourths. 136 00:10:18,520 --> 00:10:23,710 Well when we have a fraction divided by a fraction instead of dividing by a fraction we can multiply 137 00:10:23,980 --> 00:10:25,480 by its reciprocal. 138 00:10:25,510 --> 00:10:31,210 So we'll get one fourth and instead of dividing that by seven fourths we'll multiply it by the reciprocal 139 00:10:31,240 --> 00:10:32,730 four sevenths. 140 00:10:32,890 --> 00:10:37,870 When we do that we'll of course get our force here in the numerator and denominator to cancel with one 141 00:10:37,870 --> 00:10:44,270 another and we're left with a value of 1 7 which is the sum of the geometric series. 142 00:10:44,290 --> 00:10:46,150 See how quick that was to find the sum. 143 00:10:46,180 --> 00:10:48,490 Once we have values of A and are it's very easy. 144 00:10:48,520 --> 00:10:53,860 So that's how you use the geometric series test to determine whether or not a geometric series is convergent 145 00:10:53,920 --> 00:10:54,950 or divergent. 146 00:10:55,120 --> 00:11:00,040 Remember that the easiest way to determine whether or not this is in fact a geometric series is just 147 00:11:00,040 --> 00:11:06,040 to go ahead and write down the first few terms in the series and then see if you can get it into this 148 00:11:06,040 --> 00:11:06,540 form here. 149 00:11:06,550 --> 00:11:11,890 If you cannot get it into this form then it's not a geometric series what makes it a geometric series 150 00:11:12,250 --> 00:11:17,620 is this value here of R which is the constant multipole right each term. 151 00:11:17,620 --> 00:11:22,540 After this we're multiplying by another negative three fourths So we have negative three fourths to 152 00:11:22,540 --> 00:11:26,970 the first and then negative three fourths times negative three fourths negative three fourths times 153 00:11:27,220 --> 00:11:32,200 three fourths I'm saying three four so we're always just multiplying the previous term by an additional 154 00:11:32,200 --> 00:11:34,000 factor of R. 155 00:11:34,030 --> 00:11:35,690 That's what makes it a geometric series. 156 00:11:35,710 --> 00:11:39,110 If you can't get in this form it's not a geometric series. 17438